91Ó°ÊÓ

Numerical integration is a useful technique for estimating the value of definite integrals, and the Midpoint Rule is one such method.
The central idea is to approximate the area under a curve by using rectangles. Instead of evaluating the function at the left or right ends of each subinterval, the function is evaluated at the midpoint. This choice tends to balance the overestimation and underestimation from each subinterval, leading to generally accurate results.
Here's the formula for the Midpoint Rule:

• The interval \([a, b]\) is divided into \(n\) equal parts.
• The width of each part is given by \( \Delta x = \frac{b - a}{n} \).
• For the Midpoint Rule, the approximation for the integral \( \int_a^b f(x) \, dx \) is calculated as \( M_n = \Delta x \sum_{k=1}^{n} f(x_{k-1}+\frac{1}{2}\Delta x) \).

Using the Midpoint Rule for our exercise, the sum of areas of rectangles gives us an approximation \( M_{25} \approx 0.6366 \).
The result is remarkably close to the actual value, with a relative error of 0%. This makes the Midpoint Rule particularly well-suited for this problem.

The Trapezoid Rule is another effective method for estimating the area under a curve. Instead of rectangles, it uses trapezoids, which can better approximate curves.
By considering each section of the graph as a trapezoid, the method acknowledges the slope between intervals, thus potentially enhancing accuracy over simple rectangles.
The trapezoid rule formula is:

• Use \([a, b]\), divide it into \(n\) equal segments.
• Each segment's width is \( \Delta x = \frac{b - a}{n} \).
• The approximation formula is \( T_n= \frac{\Delta x}{2}\left(f(a) + 2\sum_{k=1}^{n-1} f\left(a+ k\Delta x\right) + f(b)\right) \).

In our scenario, the Trapezoid Rule offers an approximation of \( T_{25} \approx 0.6201 \).
Despite the sophistication of trapezoids, in certain cases, like here, an alternative method (i.e., Midpoint Rule) might provide a closer approximation to the exact integral.

When using numerical methods, it's crucial to understand how close an approximation is to the true value. Here comes the concept of relative error, which offers insight into the accuracy of an approximation.
Relative error gives the discrepancy between the estimated and actual values, expressed as a percentage of the real value.
It's calculated using the formula:

• For an approximation \(A\) and the true value \(T\), relative error is \( \frac{|T - A|}{|T|} imes 100\% \).

In our example, the Midpoint Rule had a relative error of 0%, meaning its result perfectly matched the actual integral value.
In contrast, the Trapezoid Rule had a relative error of approximately 2.60%, indicating a slight deviation from the true value.
Keeping an eye on relative error is critical, as it helps assess which method better suits a particular problem, ensuring accuracy and reliability in numerical calculations.